Problem: The lifespans of lizards in a particular zoo are normally distributed. The average lizard lives $3.1$ years; the standard deviation is $0.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a lizard living longer than $2.5$ years.
Solution: $3.1$ $2.5$ $3.7$ $1.9$ $4.3$ $1.3$ $4.9$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $3.1$ years. We know the standard deviation is $0.6$ years, so one standard deviation below the mean is $2.5$ years and one standard deviation above the mean is $3.7$ years. Two standard deviations below the mean is $1.9$ years and two standard deviations above the mean is $4.3$ years. Three standard deviations below the mean is $1.3$ years and three standard deviations above the mean is $4.9$ years. We are interested in the probability of a lizard living longer than $2.5$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the lizards will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the lizards will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $2.5$ years and the other half $({16\%})$ will live longer than $3.7$ years. The probability of a particular lizard living longer than $2.5$ years is ${68\%} + {16\%}$, or $84\%$.